SharedMemory: first cut at operational semantics

SharedMemory.v

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+(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
+  * Chapter 13: Operational Semantics for Shared-Memory Concurrency
+  * Author: Adam Chlipala
+  * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
+
+Require Import Frap.
+
+Set Implicit Arguments.
+Set Asymmetric Patterns.
+
+(** * Shared notations and definitions; main material starts afterward. *)
+
+Notation "m $! k" := (match m $? k with Some n => n | None => O end) (at level 30).
+Definition heap := fmap nat nat.
+Definition assertion := heap -> Prop.
+
+Hint Extern 1 (_ <= _) => linear_arithmetic.
+Hint Extern 1 (@eq nat _ _) => linear_arithmetic.
+
+Ltac simp := repeat (simplify; subst; propositional;
+                     try match goal with
+                         | [ H : ex _ |- _ ] => invert H
+                         end); try linear_arithmetic.
+
+
+(** * An object language with shared-memory concurrency *)
+
+Inductive loop_outcome acc :=
+| Done (a : acc)
+| Again (a : acc).
+
+Inductive cmd : Set -> Type :=
+| Return {result : Set} (r : result) : cmd result
+| Bind {result result'} (c1 : cmd result') (c2 : result' -> cmd result) : cmd result
+| Read (a : nat) : cmd nat
+| Write (a v : nat) : cmd unit
+| Loop {acc : Set} (init : acc) (body : acc -> cmd (loop_outcome acc)) : cmd acc
+| Fail {result} : cmd result
+
+(* Now here's the new part: parallel composition of commands. *)
+| Par (c1 c2 : cmd unit) : cmd unit
+
+(* Let's also add locking commands, where locks are named by [nat]s. *)
+| Lock (a : nat) : cmd unit
+| Unlock (a : nat) : cmd unit.
+
+Notation "x <- c1 ; c2" := (Bind c1 (fun x => c2)) (right associativity, at level 80).
+Notation "'for' x := i 'loop' c1 'done'" := (Loop i (fun x => c1)) (right associativity, at level 80).
+Infix "||" := Par.
+
+Definition locks := set nat.
+
+Inductive step : forall A, heap * locks * cmd A -> heap * locks * cmd A -> Prop :=
+| StepBindRecur : forall result result' (c1 c1' : cmd result') (c2 : result' -> cmd result) h h' l l',
+  step (h, l, c1) (h', l', c1')
+  -> step (h, l, Bind c1 c2) (h', l', Bind c1' c2)
+| StepBindProceed : forall (result result' : Set) (v : result') (c2 : result' -> cmd result) h l,
+  step (h, l, Bind (Return v) c2) (h, l, c2 v)
+
+| StepLoop : forall (acc : Set) (init : acc) (body : acc -> cmd (loop_outcome acc)) h l,
+  step (h, l, Loop init body) (h, l, o <- body init; match o with
+                                                     | Done a => Return a
+                                                     | Again a => Loop a body
+                                                     end)
+
+| StepRead : forall h l a,
+  step (h, l, Read a) (h, l, Return (h $! a))
+| StepWrite : forall h l a v,
+  step (h, l, Write a v) (h $+ (a, v), l, Return tt)
+
+| StepParRecur1 : forall h l c1 c2 h' l' c1',
+  step (h, l, c1) (h', l', c1')
+  -> step (h, l, Par c1 c2) (h', l', Par c1' c2)
+| StepParRecur2 : forall h l c1 c2 h' l' c2',
+  step (h, l, c2) (h', l', c2')
+  -> step (h, l, Par c1 c2) (h', l', Par c1 c2')
+| StepParProceed1 : forall h l c2,
+   step (h, l, Par (Return tt) c2) (h, l, c2)
+| StepParProceed2 : forall h l c1,
+   step (h, l, Par c1 (Return tt)) (h, l, c1)
+
+| StepLock : forall h l a,
+  ~a \in l
+  -> step (h, l, Lock a) (h, l \cup {a}, Return tt)
+| StepUnlock : forall h l a,
+  a \in l
+  -> step (h, l, Unlock a) (h, l \setminus {a}, Return tt).
+
+Definition trsys_of (h : heap) (l : locks) {result} (c : cmd result) := {|
+  Initial := {(h, l, c)};
+  Step := step (A := result)
+|}.